IndisputableMonolith.Foundation.PrimitiveRecognitionCalculus.Factorization.PeriodFactor
Even recognition periods admit a nontrivial algebraic factor via the difference-of-squares identity on natural numbers. The module proves the truncated-subtraction form of that identity for the second factor at least one, then packages the lift to even period gaps as an explicit certificate. Gap analysis and the factorization master certificate import this package. The argument is elementary algebra plus parity bookkeeping on the period.
claimIn $\mathbb{N}$ with truncated subtraction, if $b \ge 1$ then $(a \dot{-} b)(a+b) = a^2 \dot{-} b^2$. Every even period therefore yields a nontrivial factor of the associated period gap, recorded as a period-factor certificate used by later gap and master-certificate results.
background
In the Primitive Recognition Calculus, discrete periods label closed recognition cycles. The eight-tick octave (forcing step T7) is the distinguished period $2^3$; more generally one studies which periods can appear and how their gaps factor. The upstream PeriodSpectrum module fixes the spectral language for those periods.
This module isolates one algebraic engine: the square-difference factorization written with truncated subtraction on $\mathbb{N}$. The identity $(a\dot{-}b)(a+b)=a^2\dot{-}b^2$ holds once $b\ge 1$, so the left factor is a genuine natural-number cofactor rather than a formal integer. That is the content flagged by the module doc-comment.
Sibling declarations then specialize the identity to even periods: an even period produces a nontrivial factorization of the period gap, packaged as a named certificate object for downstream import.
proof idea
The module is a short factorization package, not a single deep theorem. First it proves the truncated square-difference identity on $\mathbb{N}$ by direct expansion and the $b\ge 1$ guard that keeps the left factor well-defined. Next it applies that identity to an even period, obtaining an explicit nontrivial cofactor of the period gap. Finally it wraps the data into a certificate structure (and a constructor lemma) so later modules can cite one object rather than rebuild the algebra. No analytic or measure-theoretic input is used.
why it matters in Recognition Science
Period factorization is the bridge from raw period spectra to structural constraints on admissible recognition cycles. This module supplies the elementary factor that EvenPeriodGap imports when it treats gaps of even periods, and that MasterCertificate imports when it assembles the global factorization certificate for the Primitive Recognition Calculus chain.
In the broader forcing picture the result supports the eight-tick story (T7): once even periods split nontrivially, the only rigid small octave compatible with the self-similar $\varphi$ fixed point and the composition law is the $2^3$ tick. Without a clean $\mathbb{N}$-level difference-of-squares lemma, those later certificates would have to redo truncated-arithmetic bookkeeping inline.
scope and limits
- Does not classify odd periods or prove they are prime in the period monoid.
- Does not derive the eight-tick octave; it only supplies a factor used toward that constraint.
- Does not address real or integer factorization outside truncated $\mathbb{N}$ arithmetic.
- Does not by itself bound physical constants, masses, or the fine-structure band.
- Does not replace PeriodSpectrum; it assumes that spectral language.